# Double jump phase transition in a soliton cellular automaton

## TL;DR

This paper analyzes a soliton cellular automaton with random initial states, revealing phase transitions in soliton sizes and uncovering a condensation phenomenon, with implications for permutation subsequences and connections to stochastic processes.

## Contribution

It provides new constructions of Young diagrams for the automaton, establishes limit theorems for soliton sizes, and uncovers a phase transition and condensation phenomena.

## Key findings

- Number of solitons scales linearly with system size n.
- Longest soliton length scales as log n, √n, or n depending on p.
- Condensation occurs in the supercritical regime for p > 1/2.

## Abstract

In this paper, we consider the soliton cellular automaton introduced in [Takahashi 1990] with a random initial configuration. We give multiple constructions of a Young diagram describing various statistics of the system in terms of familiar objects like birth-and-death chains and Galton-Watson forests. Using these ideas, we establish limit theorems showing that if the first $n$ boxes are occupied independently with probability $p\in(0,1)$, then the number of solitons is of order $n$ for all $p$, and the length of the longest soliton is of order $\log n$ for $p<1/2$, order $\sqrt{n}$ for $p=1/2$, and order $n$ for $p>1/2$. Additionally, we uncover a condensation phenomenon in the supercritical regime: For each fixed $j\geq 1$, the top $j$ soliton lengths have the same order as the longest for $p\leq 1/2$, whereas all but the longest have order at most $\log n$ for $p>1/2$. As an application, we obtain scaling limits for the lengths of the $k^{\text{th}}$ longest increasing and decreasing subsequences in a random stack-sortable permutation of length $n$ in terms of random walks and Brownian excursions.